Perception &Psychophysics1980, Vol. 27 (2),93·103

Eliminating subjective biasesin judging the loudness of a l-kHz tone

E. C. POULTON and R. S. EDWARDSMedical Research Council, Applied Psychology Unit, Cambridge CB22EF, England

and

T. J. FOWLERDepartment 01Engineering, Cambridge University, Cambridge, England

It is possible to eliminate most of the known subjective biases that affect judgments ofsensory magnitude using numbers. Experiments are described which do this, and which alsoinvestigate some of the biases. The least biased estimate for doubling the loudness of al·kHz tone is found to be about 11.5 dB. This value is still slightly affected by the logarithmicbias, although the bias could be eliminated. It is also affected by the stimulus equalizing bias,produced by the inequality between the finite range of loudnesses to which the ears aresensitive and the infinite range of numbers to which the loudnesses are matched. This lastbias cannot be eliminated completely in direct magnitude estimation.

ly than any other sensory dimension (Marks, 1974;Stevens, 1955).

Range BiasesThe composite model of Figure 2 illustrates the

relationship between the range biases, which are Iistedat the top of Table I. The figure shows two sizes ofresponse range plotted against two sizes of stimulusrange.

In the centering bias of Figure IA, the observercenters his range of responses upon the range of stim­uli. In Figure 2, the centering bias determines the

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As a rough approximation, the biases that affectquantitative subjective judgments can be separatedinto two distinct kinds. First, there are the specificbiases that are introduced by the experimenter'schoice of the experimental details. In direct magni­tude estimation, they include the choice of the exactstimuli, the responses, and the standard, if used.These specific experimenter biases are reviewed byPoulton(l968, Figure I).

The present investigation is concerned with thesecond, more general kind of biases that are intro­duced by the ob server in making his quantitativesubjective judgrnents. They are illustrated in Figure 1and listed on the left of Table I (Poulton, 1979).These more general subjective biases cannot be sepa­rated entirely from the specific experimenter biases,because experimental results depend upon both theexperimenter's detailed choices and the observer'schoices. Hut in studying subjective biases, the experi­menter can attempt to minimize the specific biasesthat he hirnself introduces, or at least to hold themconstant across comparisons.

The present paper describes an attempt to eliminatethe subjective biases introduced by the ob server. Twoof the biases are investigated separately. The right sideof Table 1 shows how the biases are dealt with. Theloudness of a l-kHz tone is selected for study becauseit has been investigated experimentally more frequent-

We are grateful to H. McRobert for permission to use his 1965data, and to R. Patterson for calibrating the volume control forus. Financial support from the Medical Research Council to thefirst two authors is also gratefully acknowledged. Requests forreprints should be addressed to E. C. Poulton, M.R.C. AppliedPsychology Unit, 15Chaucer Road, Cambridge CB2 2EF. England.

Figure I. Models for the subjective biases in judging sensorymagnitudes. Models A, H, and C are concerned with the overallrange, whereas Models D, E, and F are concerned with the non.Iinearities within Ihe overall range (8 = stimulus; R = response).(From Poulton, 1979.)

Copyright 19RO Psychonornic Socicty, lnc. 93 0031-5117/80/020093-11 $01.35/0

94 POULTON, EDWARDS, AND FOWLER

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Figure 2. A composite model of the range biases. The centeringbias determines the point at which the three straight Iines cross.The contraction bias reduces the slopes of the two solid Iines com­pared with the slope of the theoretically unbiased dashed llne,The difference in slope between the two solid Iines reßects thestimulus and response equalizing biases. The slope is steeper witha small stimulus range (s) or a large response range (R), thanwith a large stimulus range (S) or a small response range (r).

cannot be avoided completely in matehing loudnessesand numbers, because the observer tends to use thesame range of responses, either loudnesses or nurn­bers, whatever the size of the range of stimuli (Poulton,1979). The bias is partly excluded by discarding themost obviously biased points. But the remainingpoints must still be biased.

In the stimulus form of the contraction bias ofFigure 1C, the observer underestimates large stimuliand differences between stimuli, and overestimatessmall stimuli and differences. In Figure 2, the dashedsloping line represents the theoretical function that isnot affected by the contraction bias. The less steepslopes of the two solid lines are due partly to the con­traction bias and partly to the stimulus and responseequalizing biases. The contraction bias reduces theslopes because the contraction bias is larger with verylarge and very small stimuli than with stimuli ofintermediate size.

Table 1 shows that the contraction bias is avoidedin the present investigation by balancing magnitudeadjustments using a volume control against numer­ical magnitude judgments, following Stevens andGreenbaum (1966). Figure 3 gives the data ofMcRobert, Bryan, and Tempest (1965) on the veryfirst numerical judgments made by groups of unprac­ticed observers judging various ranges of intensity of

Figure 3. McRobert et al.'s (1965) very first multiple judgmentsof loudness of a l-kHz tone. The open points on the abscissarepresent the standard tone, which is always presented first andcalled 1.0. The open points on the dashed horizontal line fortwice loudness show the theoretlcal slope of 10 dU for twice loud­ness, The filled points represent the geometric means of separategroups of between 9 and 19 undergraduates or university staffwho are not practlced Iisteners. The sloping Iines connect the geo­metric means to their standards. Lines passing to the left of thecorresponding theoretical point for twice loudness indieate thatless than 10 dU is required for twice loudness, Lines passlng to theright indieate that more than 10 dU is required.

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position of the point in the middle through whichthe sloping lines pass. Parducci (1963; Parducci &Perrett, 1971) combines the centering bias with thestimulus-spacing bias of Figure 1E in his range­frequency model. The model extends and modifiesHelson's (1964) original adaptation-level model ofthe two biases. The top part of Table 1 shows thatin the present investigation, the centering bias isavoided by using the very first judgments of groupsof unpracticed observers.

The bias affecting the slopes of the lines in Figure2 can be described most conveniently by dividing itinto two parts. The contraction bias of Figure 1C isdefined as the symmetrical part, 1t corresponds toStevens and Greenbaum's (1966) regression effect.The remaining asymmetric part of the bias becomesthe stimulus and response equalizing biases of FigureIB.

In the stimulus and response equalizing biases ofFigure 1B, the ob server equates whatever the size ofthe stimulus range with whatever the size of hisresponse range. The steeper solid line, labeled sR inFigure 2, represents a small stimulus range pairedwith a large response range. The less steep solid line,labeled, Sr, represents a large stimulus range pairedwith a small response range, Table 1 shows that in thepresent investigation the response equalizing bias isavoided by leaving the observer free to choose herown range of responses. The stimulus equalizing bias

ELIMINATING SUBJECTIVE BIASES 95

Table ISubjective Biases in Judging Sensory Magnitudes

Bias

I. Range BiasesCentering

The observer centers his range of responses uponthe range of stimuli

Stimulus EqualizingWhatever the size of the stimulus range, the ob server

- uses his full range of responsesResponse Equalizing

Whatever size response range the observer is given, hedistributes the responses over the stimulus range

ContractionThe observer underestimates large stimuli and differ­ences between stimuli and overestimates small stimuliand differencesOnce the observer knows the range of responses, heselects a response too close to the middle of the range

2. Nonlinear BiasesLocal Contraction

The observer treats stimuli in small very high intensityand very low intensity ranges as if they have lessextreme values

Stimulus SpacingThe observer responds as if the stimuli are equallyspaced geometrically and equally probable

LogarithmicWhen there is a step change in the number of digits,the observer compromises between a pure linear anda pure logarithmic scale

3. Transfer BiasesTransfer from Previous InvestigationsTransfer from Instructions and DemonstrationsTransfer from Previous StimuliTransfer from Previous JudgmentsTransfer from Previous Responses

a l-kHz tone. The present investigation providessome of the corresponding data when the observer isgiven a volume control and is told to set it to an inten­sity a specified number of times greater than a stan­dard intensity. Balancing the data obtained by thetwo methods eliminates the symmetrical contractionbias, because, by definition, the contraction biases ofthe two methods are equal and opposite in direction.The remaining assymmetries between loudness andnumbers, produced by the difference in the sizes ofthe ranges, are, by definition, the stimulus andresponse equalizing biases.

As indicated at the bottom of Part 1 of Table I,the contraction bias has a response form that occurswhen the observer is given a range of responses with aknown middle value. The observer selects a responsetoo elose to the middle of the range. In the presentinvestigation, this means being familiar with the vol­urne controI. It is avoided by providing an unknownvolume control for the very first judgment. The

How Dealt With

Avoided by using the very first judgment of unpracticedobservers

Partly excluded by discarding the most biased medianjudgments

Avoided by leaving the observer free to choose the sizeof his range of responses .

Avoided by balancing magnitude adjustments againstmagnitude estimates

Investigated separately, Avoided by using a volume con­trol with an unknown center and a numerical responserange that can extend from 1.0 to infinity without anobvious middle value

Investigated separately. Avoided by not using a smallstimulus range with extreme values

Avoided by using the very first judgment

Avoided for the gain adjustments by using unpracticedobservers and single-digit numbers between land 10.Bias-free data not available for the numerical judgments

Investigated separately

Avoided by using the very first judgment of unpracticedobservers with unbiased instructions and no demonstra­tions

effect of seeing the volume control and its calibra­tions before using it is examined separately. In theexperiment of Figure 3, MeRobert et aI. (1965) avoidthis difficulty of a response range with an knownmiddle value by asking for multiple numerical judg­ments. Here the responses range from 1.0 to infinity,and so have no obvious middle value.

Nonlinear BiasesThe composite model of Figure 4 illustrates the rela­

tionships between the nonlinear biases, which are listedin Part 2 of Table 1. The figure shows a dashed fune­tion, whose center seetion is relatively free of bias,and two biased solid functions. Numerieal responsesare given on a linear seale on the ordinate. Stimulusintensity is given on a logarithmie scale on theabscissa.

The loeal eontraetion bias of Figure 10 is a non­linearity that is deseribed so far only for judgmentsof loudness (Poulton & Stevens, 1955). The ob server

96 POULTON, EDWARDS, AND FOWLER

Figure 4. A composite model of tbe nonIinear biases. Tbe lowersolid function Illustrates tbe logaritbmic blas. Tbe upper solidfunctlon illustrates tbe stimulus spaclng blas in tbe tbeoreticalabsence of tbe logaritbmlc blas. Tbe dasbed functlon sbows tbestimulus spacing bias comblned witb and partly cancelllng tbelogaritbmic bias. Tbe local contraction blas alters tbe slope oftbe dasbed function at its top and bottom ends (see text),

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there is a step change in the number of digits usedas stimuli or responses. To an observer who uses alogarithmic scale, there are as many numbers between10 and 100 as there are between 1 and 10, whereasto an observer who uses a linear scale, there are 10times as many numbers between 10 and 100 as thereare between 1 and 10. Untrained ob servers do notnormally use numbers in either of these two ways.They compromise between the two pure scales, butuse a scale that is nearer to logarithmie than to linear(Pou1ton, 1979).

In making numerical judgments, the logarithmiebias can affect the means, as well as the individualjudgments. Since logarithmic and linear scales arerelated to each other nonlinearly, the observer's com­promise scale is related nonlinearly to both scales.This biases both arithmetie means, whieh assurne alinear scale, and geometrie means, whieh assurne alogarithmie scale. Only medians and nonparametriestatisties are not affected by the nonlinearity pro­duced by the logarithmie bias.

Figure 1F shows that, compared with a linear scale,the logarithmie bias shrinks the upper part of thenumerieal scale. The lower solid function in Figure 4illustrates the logarithmie bias in aseries of judg­ments. The dashed function shows how the stimulusspacing bias can be made to cancel the logarithmicbias over the middle of the range of stimuli. Table 1shows that the logarithmie bias is avoided by usingunpracticed observers and single-digit numbers between1 and 10.

Range and Nonlinear BiasesFigure 5 illustrates the distribution of the observer's

responses that will account for all the range and non­linear biases of Figure 1 and Table 1. The modelmakes no assumptions about the size of the range ofstimuli se1ected by the experimenter. Any sized rangewill do. The symmetrie distribution of responses pro­duces the centering bias. Using almost the wholerange of responses, whatever the size of the stimulus

Numerical response range

Figure 5. The tbeoretical distribution of tbe observer's responsestbat will aeeount for all tbe range and nonlinear blases Iisted InTable 1 (see text), Tbe numerical scale on tbe absclssa representstbe observers' responses In magnitude estimation. But tbe modelworks equally weil wben tbe experimenter presents tbe numbersand tbe observer adJusts tbe position of a volume control to matcbthem.

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treats acoustic stimuli in small very high-intensityand very low-intensity ranges as if they have lessextreme values. For multiple judgments of high inten­sity, it reduces the slope of the dashed function inFigure 4 at the top end. But for fractional judg­ments of high intensity, it increases the slope, At thebottom 'end of the dashed function for judgments oflow intensity, the local contraction bias has the reverseeffect. It increases the slope for multiple judgments,but reduces the slope for fractional judgments. Part 2of Table 1 shows that the local contraction bias is pre­vented by avoiding a small stimulus range with extremevalues. This is one of the biases that is examined sepa­rately in the present investigation.

In the stimulus spacing bias of Figure 1E, the obser­ver responds as if all the stimuli are equally spacedgeometrically and equally probable. Equal geometriespacing means equal intervals on a logarithmie scalelike that on the abscissa of Figure 4. The unfilled circleson the top function represent stimuli that are moreclosely spaced horizontallyon the 1eft of the figurethan on the right. In the theoretical absence of the loga­rithmic bias, the observer allocates numbers to thestimuli as if they are equally spaced, judging each stim­ulus to be separated from the next by 50. Thus, thesolid function is steeper on the left than on the right.Table 1 shows that the stimulus spacing bias is avoidedin the present investigation by using very first judg­ments.

The logarithmic bias of Figure 1F occurs whenever

range, produces the stimulus and response equalizingbiases. Avoiding the extreme responses, using insteadless extreme values, produces the contraction biases.The flat top to the distribution indicates that theobserver uses all except the extreme responses aboutequally often. This produces the stimulus spacingbias. Giving I-digit, 2-digit, and in general n-digitnumbers alm ost equal distances on the numericalscale of the abscissa produces the logarithmic bias.

Transfer BiasesThe bottom part of Table I lists the transfer biases.

They are avoided in the present investigation by usingthe very first judgments of unpracticed ob servers ,with unbiased instructions and no demonstrations.The transfer bias from one trial to the next is one ofthe biases that is examined here.

METHOD

ApparatusTwo response buttons are mounted on a horizontal panel c1amped

to the side of a table. Pressing the white left-hand button producesa l-kHz tone of 39 dB re 20 I'N/m' in the headphones worn by theobserver. Pressing the red right-hand button produces a l-kHztone whose intensity depends upon the setting of a volurne control.

The volume control is mounted on an instrument panel on theobserver's right. The instrument panel rests on the edge of thetable, facing away from the observer. the observer reaches the vol­urnecontrol by pushing her right hand under a black curtain whiehprevents her from seeing the instrument panel. She rests her fore­arm on the table with her elbow bent to about a right angle.

The knob of the volume control has a diameter of 5.5 cm androtates through 315 deg. 11 has a circular 15-cm-diam disk of clearcelluloid attached to its back, a semicircle of which is obscured byblack paper. In Experiment I, where the observer is shown the vol­urne control, the black paper obscures half its calibrations. Theobserver can see only the numbers I through 13, arranged evenlyround the semieircle of the dial that is visible.

The response characteristie of the volume control is illustrated inFigure 6. It resembles fairly close1y the "sone potentiometer" usedby Stevens and Poulton (1956, Figure 1). But it deviates a little inthe direction of a more linear relationship between the angle ofrotation and decibels. Only one observer ever approaches the topintensity of 115 dB. Before each trial, the volume control is set tothe intensity of the standard tone of 39 dB.

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ELIMINATING SUBJECTIVE BIASES 97

ProcedureFirst, the observer is asked whether she has ever judged loud­

ness before. In Experiment I, she is then shown the volume controlthat she will be using. In Experiments 2 and 3, the volume control isalways kept out of sight. When putting on the headphones, theobserver is asked to make sure that they fit comfortably and toremove any hair that gets in the way.

The instructions run as folIows: "When you press the left-handwhite key, you will hear a soft tone. We can call its loudness one.Always press keys for a second or two. (The observer is instruct­ed to press the key.) When you press the right-hand red key, theloudness of the tone depends upon how far up you have turned thevolume control behind the black material. (The experimenterplaces the observer's right hand on the volume control.) I want youto adjust the volume control so that the loudness of the right-handtone is twice (10, 30, or 100 times) the loudness of the soft tone onthe left. When you compare the two loudnesses, press the left keyfor a second or two, then pause, then press the right key for asecond or two. You can compare the two loudnesses as often asyou like, until you have set the volume control to what sounds likethe correct loudness. When you are satisfied with your choice,please let me know. I suggest that you listen to both louder andsofter tones from the volume control before you make yourchoiee."

Any questions are answered by paraphrasing the instructions. Ifnecessary, it is emphasized that the experimenter does not knowappropriate setting of the volume control, This is what the experi­ment is designed to find out.

Experimental DesignIn Experiments land 2, each observer makes four adjustments

to, respectively, twiee, 10, 30, and 100 times the loudness of thestandard. The order of the conditions is shown in Table 3. InExperiment I, there are 9 observers in each group. In Experiment2, there are 12or 13observers in each group, as indicated in Table 3.

In Experiment 3, each observer makes two adjustments, to 6and 36 times the loudness of the standard, respectively. One groupof 23 observers makes the two adjustments in this order. The othergroup of 22 observers makes the adjustments in the reverse order.

SubjectsWomen members of the MRC Applied Psychology Unit subject

panel served in Experiments land 2, 36 and 49 of them, respec­tively, Their ages ranged from 19 to 55 years, median age 44. Theyperformed the experiment on arrival, before serving in an experi­ment on visual search. They were paid for their services. Four hadserved about 9 years previously in an experiment estimating themultiple loudness of a narrow band of noise centered on 1kHz(Poulton, 1969).

The 45 unpaid volunteers in Experiment 3 comprised 39 menand 6 women. They were caught as they passed through thefoyer of the Cambridge University Engineering Department. Mostwere undergraduates of the department. Their ages ranged from 18to 43 years, median age 22. None said they had judged loudnessespreviously. In all three experiments, the people were allocated toconditions in their order or appearance. The only exceptions werethe four women who reported that they had judged loudnesspreviously. They were distributed evenly between the main groups .

Caiculations and Statistical TestsIn any experiment investigating an unknown, perhaps non­

linear, relationship between an independent and adependentvariable, medians and nonparametric statistieal tests are the obvi­ous choiees. However, many experimenters calculate means indecibels, whieh are a logarithmie transformation of energy. Bothmedians and means are therefore given in Figures 7 and 8 forthe gain adjustments.

For the numerical judgments, geometric means are used. Unfor­tunately, McRobert et al. 's (1%5) data are no longer avalable forcalculating medians. However, some indieation of the Iike1y dif­ference between the geometric means and the medians of the very

98 POULTON, EDWARDS, AND FOWLER

Figure 7. Very Ilrst multiple judgments 01 loudness by separategroups 01unpractleed observers. Tbe dasbed IInesbows the pooledgaln adjustments 01 Experiments 1 and 2. Tbe unIllIed squaresrepresent tbe medlans. The unIllIed c1rcles represent the cerre­spondlng means In declbels. Eaeh median or mean Is lor a sepa­rate group 01 21 or 22 women. The solid line shows tbe geo­metrieally pooled geometrie mean numerieal judgments 01 Fig­ure 3 (MeRobert et al., 1965). Eaeh filled elrele represents a sepa­rate group 01 at least 50 undergraduates, except lor the fIIledc1rcle on tbe rlgbt, whleb Is lor only 24 undergraduates.

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RESULTS

first multiple numerieal judgments of loudness ean be obtainedfrom the data of Poulton (1969), who used an oetave band ofnoise eentered on 1kHz and about 30 housewives in eaeh group.Here geometrie means of 2.3, 7.2, 8.0, and 28.4 eorrespond tomedians of 2,6, 7, and 20, respeetively. Thus, the logarithmie biashas a mueh greater effeet upon very first judgments with geo­metrie means greater than 10 than with geometrie means less than10, as is to be expeeted. Only nonparametrie statistieal tests areused, Mann-Whitney U tests and Wilcoxon tests, All tests are two­tailed.

Pooled BissesFigure 7 gives the pooled results of Experiments 1

and 2 as open points. The open squares represent themedians of 21 or 22 very first adjustments. The opencircles represent the eorresponding means in decibels.The filled eircles give the data from Figure 3 (MeRobertet al., 1965). They represent the geometrieally pooledgeometrie means of at least 50 very first numeriealjudgments, exeept for the filled circle on the right foronly 24 judgments.

In all the experiments, the standard is shown at theorigin with an intensity of 0 dB and a numerieal val­ue of 1. The abscissa shows the inerease in intensityabove the standard. Thus, all the standards on theabscissa of Figure 3 are superimposed in Figure 7.The ordinate shows subjeetive loudness on a loga­rithmie seale.

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Figure I. Tbe least blased very fint multiple judgments 01 loud­neu. Tbe unlUied squares represent the medlans 01 Experiment 2,mtbout prior vlew 01 the volume eontrol and wltbout tbe twoobserven wbo judged loudneu I} yean prevlously. Tbe unIllIedtrIangles represent the medlans 01 Experiment 3. Tbe unfilled elr­eies show tlle eorrespondlng means In declbels. All values areInereased by 1.0 dB to super1mpose tbe 31}-dB standard upontbe 4O-dB standard used by MeRobert et al. (1965) lor tbe IlIIedelreles. The unIllIed points In braekets are greatly blased. Tbedasbed IIne eonnects the two least blased medlans. The solid IIneIs fitted to tbe slx IlIIed elreles wbleb eaeb represent the geometriemean 01 between I} and 15 observers. The dotted and dashed IIneeonnects the orlgln to the point where the other two IInescross.

Contraction BiasThe results in Figure 7 look reasonable orderly for

very first judgments, beeause eaeh point represents atleast 21 people, The erossing of the broken funetionfor the mean in decibels of the gain adjustments bythe solid funetion for the geometrie means of thenumerical judgments, represents the eontraetion bias.It eorresponds to Stevens and Greenbaum's (1966)regression effeet. The erossing point is the only pointthat is by definition not affeeted by the eontraetion

.bias. It represents a loudness ratio of 8 for a differeneeof 37 dB from the standard. With the logarithmieseale on the ordinate, this means about 12 dB fortwiee or half loudness. It gives an exponent for soundamplitude of .5. (The slopes on a log log plot likethat of Figure 7 have to be doubled to give Stevens'exponents.)

The value of 12 dB is a little above Marks' (1974,Figure 1) modal value of about 11 dB reported in 23experiments. It is a little further still above Stevens'(1955) eomposite value of about 10 dB for twiee orhalf loudness, whieh was based upon all the previouslypublished data at that time.

Stimulus Equalizing BissOne unusual eharaeteristie in Figure 7 is the almost

vertical slope of the dashed funetion for the gain

ELIMINATING SUBJECTIVE BIASES 99

Table 2Effect of Seeing the Volume Control Before Making

the Very First Adjustment

that 65 dB represents a rotation of only about 10%of the total range of rotation of the volume control.Thus, although seeing the volume control biases themedian in the direction of the center of its range ofmovement, the median setting is nowhere near themiddle of the range.

The largest median in Table 2 is 53 dB, for anincrease in loudness of x 100. Adding the 39 dB of thestandard gives a median intensity of 92 dB. Figure 5shows that even this represents a rotation of only40070 of the total range of rotation of the volume con­trol. Thus, in spite of seeing the volume control,all the medians are on the lower side of the center ofits range of rotation, like the intensity correspondingto the standard.

Local Contraction BiasThe local contraction bias is described only for

acoustic stimuli in small very high- and very low-inten­sity ranges. In Table 2, it applies only to the gainadjustment x 2. The local contraction bias increasesthe number of decibels required to double the intensityof a soft sound, compared with the number requiredto double the intenstiy of asound of moderate inten­sity. Here the bias is in the same direction as thecontraction bias, which accounts for the overesti­mation of small differences. For the observers inExperiment 1, the bias is also in the same direction asthe volume control contraction bias. Thus, the resultsof Experiment 1 need to be excluded in attempting toisolate the local contraction bias.

Table 2 shows that, in Experiment 2, twice theloudness of the standard of 39 dB requires a medianincrease in intensitv of 17 dB. The value of 17 dBis reliably (p < .05) larger than Stevens' (1955) com­posite value of 10 dB for twice or half the loudness oftones and Marks' (1974) modal value of 11 dB fortwice or half loudness. The relatively large value of17 dB is increased by the contraction bias, whichincreases all judgments of small differences like twiceor half loudness, as weIl as by the local contractionbias. However, if anything, Stevens' and Marks' val­ues of 10 and 11 dB are also likely to be slightlyincreased by the contraction bias.

Stevens' (1955) composite value of 10 dB is basedlargely upon judgments of halving or doubling. Thus,they correspond most closely to the points on the

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adjustments at the top on the right. For the mediansrepresented by the unfilled squares, the top segment isvertical. The median gain adjustment is the same sizewhether the women are adjusting to 30 times the loud­ness of the standard or to 100 times the loudness.The women appear to be running out of responses.This is a ceiling effect imposed by the stimulus equal­izing bias.

In obtaining gain adjustments, the experimenter hasas infinite range of numerical stimulus ratios thatextends from 1.0 to infinity, whereas the observer'sresponse range of physical intensities is limited bywhat the ears can withstand or by what the observeris willing to inflict upon her ears. This corresponds tothe less steep function Sr in Figure 2 for a largestimulus range paired to a small response range. InFigure 7, for the gain adjustments, the stimuli areplotted vertically while the responses are plotted hori­zontaIly. This is the opposite of the labeling of theaxes in Figure 2. Thus, the almost horizontal line Srin Figure 2 corresponds to the almost vertical dashedline segments in Figure 7.

The solid function in Figure 7 for the numericaljudgments must also be too steep. In obtaining numer­ical judgments, the sizes of the stimulus and responseranges are reversed. Here the experimenter has a finiterange of stimulus intensities, while the observer hasinfinite range of numerical ratios. This correspondsto the steeper function sR in Figure 2 for a smallstimulus range paired to a large response range. Thebias increases the steepness of the solid function inFigure 7 for the numerical judgments. However, theright-hand segment of the solid line is not as steepas the final segment of the dashed line, owing to thecontraction bias which has already been discussed.

Volume Control Contraction BiasThe response form of the contraction bias, at the

bottom of Part 1 of Table I, occurs when the rangeof responses has an obvious middle value and theobserver knows the size of the range. In magnitudeadjustments, this can happen when the observer knowsthe range of movement of the volume control. Beforethey start the experiment, the observers in Experiment1 are shown the volume control and its calibrations inthe arbitrary units of distance given on the left ordinateof Figure 6, while the observers in Experiment 2 arenot. The effect upon the median difference from thestandard of the very first adjustments is shown inTable 2.

The median differences from the standard of thegroups of observers in Experiment 1, who see the vol­urnecontrol, are alilarger than the corresponding medi­ans of the groups of observers in Experiment 2, who donot. For the smallest angle of rotation, to increaseloudness x 2, the increase in decibels from 17 to 26is reliable (p < .02).

A median of 26 dB above the standard of 39 dBmakes a median intensity of 65 dB. Figure 6 shows

100 POULTON, EDWARDS, AND FOWLER

extreme left of the two functions of Figure 7. Con­necting these two points to the standard at the originshows that the gain adjustments have the smallerslope, and hence require more decibels for doublingand halving than do the numerical judgments. Stevensincludes in his data ab out twice as many gain adjust­ments as numerical judgments. Thus, if anything, hiscomposite value of 10 dB should be slightly too large.

This argument applies also to the seven experimentson halving or doubling used by Marks (1974). Here,again, there are about twice as many gain adjust­ments as numerical judgments. However, Marks'modal of value of 11 dB is based predominantly uponthe slopes of complete psychophysical functions likethose of Figure 7. Here the figure shows that it is thenumerical judgrnents that have the smaller slope, andhence require more decibels for doubling and halvingthan do the gain adjustments. Marks includes aboutthree times as many complete psychophysical func­tions derived from numerical judgments as from gainadjustments. The predominance of complete psycho­physical functions derived from numerical judgments,which require more decibels for doubling and halving,means that his modal value of 11 dB should be slight­ly too large, like Stevens' composite value of 10 dB.

In comparing the 17 dB required for doublingloudness of Experiment 2 in Table 2 with the valuesof 10 and 11 dB given by Stevens and by Marks,all three values are therefore likely to be exaggeratedby the contraction bias, although the 17-dB valuemay be the most exaggerated. Thus, much of thereliable difference between the 17 dB and the valuesof 10 and 11 dB is likely to be due to the local con­traction bias, although some of it is probably dueto the contraction bias.

Bias by TransferTable 3 shows the effect of prior gain adjustments in

Experiment 2. The orders of the four trials performedby the four groups of observers are shown in the fourcolumns on the left. The median differences from thestandard in the gain adjustments are shown in the fourcolumns on the right,

The first row of Table 3 on the right shows themedian differences from the standard of the very firstgain adjustments. They correspond to the values inrow 2 of Table 2. The medians for x 10, x 30, andx 100 are all about the same, 38 to 40 dB. The sub­sequent rows of the table show that the medians aremore spread out in the later trials. The medians forx 10 fall reliably from 38 dB in the first trial to 30 dBin the second trial (p < .002), and from then onremain at 30 dB or below. The medians for x 30oscillate in Trials 2 to 4 around the 39 dB of Trial 1,while the medians for x 100 increase reliably from40 dB in the first trial to 49 dB in the second trial(p < .05) and then increase gradually to 53 dB in thefourth trial. The spreading out reflects the influenceof the previous judgments in the absence of any

Table 3Effect of Transfer from Preceding Trials in Experiment 2

Median Increase in

OrderDecibels for

ofCon-N of Observers SL Ratio

ditions 13 12 12 12 X2 XI0 X30 X100

1 X2 XlO X30 XI00 17 38* 39 40t2 XI0 X100 X2 X30 19 30 30 49t3 X30 X2 XI00 XI0 17 24 40 504 XI00 X30 XlO X2 18 30 37 53

Note- Volume control not seen. SL = subjective loudness.·XlO first largest in column (p < .05 or better)fX100 first sma/ler than X100 second (p < .05)

external knowledge of results. It shows how thestimulus spacing bias of Figure lE operates over aseries of magnitude judgments.

The spreading out does not apply to twice loud­ness. Here Table 3 shows that the median differencefrom the standard remains throughout at about 17 dB.Most of the early investigations of judgments ofloudness stick to ratios of twice and half loudness(Stevens, 1955). The results in Table 3 suggest thatthis avoids the transfer that may bias the larger ratiosof loudness when they are included in a within­subjects design.

Four women judged the loudness of a narrow bandof noise centered on 1 kHz about 9 years beforeserving in the present experiments. The standard wasabout 65 dB, and the variables ranged from 70 to100 dB (poulton, 1969). One woman served in each ofthe four composite groups represented by the openpoints in Figure 7. Two of the women made adjust­ments above the median of their group, while twomade adjustments below the median. Their resultsbear no obvious relationship to their experience 9years previously.

Logarithmic BiasThe logarithmic bias in the use of numbers pro­

duces a number scale that is intermediate betweenlinear and logarithmie, as illustrated in Figure IF.On the log log plot of Figure 7, it produces functionsthat are concave downward, though not as concavedownward as a linear number scale. The direction ofthe bias depends upon whether it is believed that theobservers should be using a linear or a logarithmienumber scale. Figure IF shows that all three numberscales are fairly similar in the region between 2 and 10.Thus, of the unfilled points for the gain adjustmentsin Figure 7, only those for x30 and x 100 are likelyto be much affected by the logarithmie bias.

Of the filled points for the numerical judgments,the point at the top on the right with a geometricmean of 25.9 is likely to be the most biased. But allthe geometrie means are Iikely to be higher than thecorresponding medians. Estimating from the data ofPoulton (1969), given in the section 011 calculations,the geometrie means of 3.1, 4.0, 6.3, 9.1, and 25.9

in Figure 7 probably represent medians of 2, 3, 5, 8,and 20, respeetively.

Minimum BiasFigure 8 illustrates an attempt to eliminate as

many biases as possible, Tbe six filled eircles are thegeometrie means of the very first numerieal judg­ments illustrated in Figure 3 for the standard of40 dB. The unfilled squares represent the medianfirst gain adjustments of Experiment 2. The unfilledtriangles represent the eorresponding medians fromExperiment 3. These are the two experiments in whiehthe observer's very first judgment is not biased by see­ing the volume eontrol before she or he starts. Notincluded are the adjustments of the two observers wbojudged loudnessabout 9 years previously.

The unfilled circles in Figure 8 represent the eor­responding means in decibels of Experiments 2 and 3.The means eorrespond to the geometrie means repre­sented by the fiUed circles for the numerical judgments.Sinee, in Experiments 2 and 3, the standard turns out tobe 39 dB, instead of the 40 dB sbown in the figure,aU the decibel values of Experiments 2 and 3 areinereased by 1.0 to make them eomparable to tbedecibel values of the filled circles.

The points in Figure 8 tbat are obviously biasedare shown in braekets. The two unfilled squares andtbe unfilled triangle all at the top, representing themedian loudness ratios of 30, 36, and 100, lie almostvertieally above the unfilled square for the medianloudness ratio of 10. Thus they are greatly affeetedby the stimulus equalizing bias. The unfilledsquare for the median loudness ratio of 2 is affeetedby the local eontraetion bias. This leaves unbraeketedonly the unfilled triangle and square for the medianloudness ratios of 6 and 10.

Transfer from previous judgments of loudness isexcluded by using only the very first judgments ofunpraetieed observers. The two women who served inExperiment 2 after serving 9 years previously in an ex­periment on numerieal judgments of loudness arealso excluded.

The unfilled triangle and square in Figure 8 for themedian very first gain adjustments to 6 and 10 timesloudness are not likely to be affeeted by the loga­rithmie bias beeause the ratios He within or at thelimits of the range of single-digit numbers. Also, theyare probably not much affeeted by the eontraetionbias, beeause they lie near the middle of the rangeof sound intensities eommonly met in everyday life,of from 60 to 80 dB. Thus they are relatively unbi­ased. When the midpoint of tbe dasbed line connectingthe two medians is joined to the standard at theorigin of the figure, it has a slope that gives about11 dB for twiee or half loudness. This is about Marks'(1974, Figure 1) modal value reported in 23 experi­ments, whieh has already been r.eferred to.

The solid line in Figure 8 is fitted to the six filled

ELIMINATING SUBJECTIVE BIASES 101

circles, whieh represent the geometrie means of thevery first numerieal judgments, When the midpointof the solid line at 75 dB and 7.6 times loudness isjoined to tbe standard at the origin of the figure, ithas a slope tbat gives about 12 dB for twiee or halfloudness.

As already pointed out, the slope would be smallerif the less biased medians were available for use. Thepairs of geometrie means for 70 and 80 dB ean eachbe eombined geometrically. Tbis gives tbe four geo­metrie means for the numerical judgments at 60, 70,80, and 90 dB, respeetively, or 3.4, 6.2, 7.5, and32.5. The data of Poulton (1969), given in the seetionon ca1culations, suggest that the correspondingmedians are most probably, 2, 5, 6, and 20. Tbesefour estimated medians ean be plotted on a graph likethat of Figure 8. When a straight line Is fitted to thefour points, its midpoint at 75 dB represents 5.9times loudness. Joining this midpoint to the standardand at the origin gives a slope tbat corresponds toabout 14 dB for twiee or half loudness. It suggeststhat the value of about 12 dB obtained from the geo­metrie means is about 2 dB too smalI.

To remove the eontraction bias, the average fortwice or half loudness of about 12 dBobtained fromthe geometrie mean numerical judgments can be bal­anced against the average of about 11 dB obtainedfrom the least biased median gain adjustments. Tbeeombined average gives about 11.5 dB for twiee orhalf loudness. Tbe dashed and dotted line in Figure 7eonnects the origin of the figure to the point wberetbe extrapolated dasbed line crosses the solid line. Italso gives about 11.5 dB for twiee or half loudness.

Unfortunately, the filled circles representing thegeometric mean numerical judgments are influencedby the logarithmie bias. Tbus, balancing the gainadjustments against tbe magnitude judgments removes

. the eontraction bias at the expense of introducingsome logarithmie bias. However, the principal biasof the dasbed and dotted line is likely to be theresidual stimulus equalizing bias, whicb is producedby the unequal sizes of the range of intensities andtbe range of numbers tbat are available. As alreadyindicated, the stimulus equalizing bias increases thesteepness of any function relating subjective magni­tude in numbers to physical intensity on a log logplot like that of Figure 8, wben tbe range of numbersavailable is infinite but the range of intensities isfinite. The bias cannot be eliminated completely inan experiment on direct magnitude estimation.

DISCUSSION OF PREVIQUS WORK

Stevens (1956) is the first experimenter to publisha paper on the biases in direct magnitude estimation.Of tbe general biases introduced by the observer , hedescribes tbe response equalizing bias of Figure 1Bthe loeal eontraction bias of Figure ID, the stimulus

102 POULTON, EDWARDS, AND FOWLER

spacing bias of Figure 1E, the logarithmic bias ofFigure IF, and bias from transfer.

Stevens (1956) also discusses some of the specificbiases introduced by the experimenter's choice of theexperimental details. He compares magnitude esti­mates using standards of different intensities and atdifferent positions in the range of variables, and withdifferent numerical values or moduli. He also reportssome results witbout a standard. He suggests (p. 14)how the standard could perhaps be changed du ringthe course of an investigation, by using a modulusfor the new standard consistent with the previousjudgments. He concludes that his least biased resultsare consistent with bis (Stevens, 1955) composite valueof 10 dB for twice or half loudness.

Hellman and Zwislocki (1961) follow up Stevens'work on the specific biases introduced by the experi­menter's choice of the experimental details, but theydo not deal explicitly with the general biases intro­duced by the observer. They combine the resultsfrom two separate ranges of intensities, a more intenserange of from 20 to 100 dB SL using a standard of70 dB, called 10, and a less intense range of fromthreshold to 60 dB SL using a standard of 40 dB,called 1. Their final psychophysical function for al-kHz tone gives about 11 dB for twice or half loud­ness over most of the range of usable intensity. Bymeasuring intensity from each person 's auditorythreshold, using sound level (SL) instead of the con­ventional sound pressure level (SPL), they are able toextend the psychophysical function for loudness downto 4 dB above threshold. Here the psychophysicalfunction becomes steeper because, of course, on alog log plot, the slope has to increase to infinity atthe threshold (Poulton, 1968, Figure 1B).

Both Hellman and Zwislocki (1961) and Stevens(1956) appear to deal with the biases intuitively, reject­ing experimental procedures because they give unac­ceptable results, rather than because they are known tointroduce bias. This is presumably the only possibleway to proceed before the biases have been de­scribed and studied. But it is probably not necessary touse this intuitive approach today, now that the biasesare better known.

In a later review, Stevens (1971) discusses the threeremaining general subjective biases of Figure 1 intro­duced by the observer: the centering bias of Figure IA,which Stevens (p. 428) eliminates by modulus equal­ization; the stimulus equalizing bias of Figure 1Bdescribed by Jones and Woskow (1962); and the con­traction bias of Figure 1C described by Stevens andPoulton (1956) and subsequently investigated morefully by Stevens and Greenbaum (1966). In 1971,Stevens proposes (p, 429) about 9 dB for twice orhalf loudness.

Warren (1970, 1973) uses the very first fractionalnumerical judgments between 100 and 0 of separate

groups of students for each intensity of the variable,holding the standard intensity constant. He avoidsthe response form of the contraction bias listed at thebottom of Part 1 of Table 1, selecting a response tooclose to the middle of the range or responses. This isdone by using the judgments only of the variablestimulus which gives a median numerical estimate of50. This variable is, on average, at the center of therange of responses between 100 and 0, where theresponse contraction bias is equal and opposite in theup and down directions. Using this method, Warrenfinds that a reduction of 6 dB produces half loud­ness as predicted by the inverse square law for thereception of energy at a distance (Warren, Sersen, &Pores, 1958).

But, unfortunately, this technique fails to avoidthe stimulus form of the contraction bias, overesti­mating the size of small stimulus differences. Injudging loudness, a difference of 6 dB is a relativelysmall difference which people tend to overestimate,like the difference of 10 dB represented by the filledcircle on the left of Figure 7. Thus, 6 dB gives toosteep a slope on a log log plot, and hence too fewdecibels for half or twice loudness.

REFERENCES

HELLMAN, R. P., & ZWISLOCKI, J. Some factors affecting theestimation of loudness. Journal 0/ the Acoustical Society 0/America, 1961,33,687-694.

HELSON, H. Adaptation-level theory, New York: Harper & Row,1964.

JONES, F. N., & WOSKOW, M. J. On the relationship betweenestimates of magnitude of loudness and pitch, American Jour­nal 0/Psychology, 1962,75,669-671.

McRoBERT, H., BRYAN, M. E., & TEMPEST, W. Magnitudeestimation of loudness. Journal 0/ Sound and Vibration, 1965,2,391-401.

MARKS, L. On scales of sensation: Prolegomena to any futurepsychophysics that will be able to come forth as science. Percep­tion 11 Psychophysics, 1974,16,368-376.

PARVUCCI, A. Range-frequency compromise in judgrnent.Psychologicat Monographs, 1963,77(2, Whole No. 565).

PARDUCCI, A., & PERRETT, L. F. Category rating scales:Effectsof relative spacing and frequency of stimulus values. Journal 0/Experimental Psychology Monograph, 1971, 89, 427-452.

POULTON, E. C. The new psychophysics: Six models for magni­tude estimation. Psychological Bulletin, 1968,69,1-19.

POULTON, E. C. Choice of first variables for single and repeatedmultiple estimates of loudness. Journal 0/ Experimental Psy­chology, 1969,80,249-253.

POULTON, E. C. Models for the biases in judging sensory magni­tudes. Psychological Bulletin, 1979,86,777-803.

POULTON, E. C., & STEVENS, S. S. On the halving and doublingof the loudness of white noise. Journal 0/ the Acoustical Society01America, 1955,27,329-331. .

STEVENS, S. S. The measurement of loudness, Journal 0/ theAcousticalSociety0/America, 1955,27,815-829.

STEVENS, S. S. The direct estimation of sensory magnitudes­loudness. American Journal 0/Psychology; 1956, 79, 1-25.

STEVENS, S. S. Issues in psychophysical measurement. Psycho­logical Review, 1971,78,426-450.

STEVENS, S. S., & GREENBAUM, H. B. Regression effect in

psychophysical judgment. Perception & Psychophysics, 1966,1,439-446.

STEVENS, S. S., & POULTON, E. C. The estimation of loudnessby unpracticed observers. Journal 0/ Experimental Psychology,1956,51,71-78.

WARREN. R. M. Elimination of biases in loudness judgments fortones. Journal 0/ the Acoustical Society 0/ America, 1970,48,1397-1403.

ELIMINATING SUBJECTIVE BIASES 103

WARREN, R. M. Quantification of loudness. American Journal 0/Psychology, 1973,86,807-825.

WARREN, R. M., SERSEN, E., & PORES, E. A basis for loudnessjudgments. American Journal 0/ Psychology, 1958,71,700-709.

(Received for publication January 2, 1979;revision accepted November 7, 1979.)